4
$\begingroup$

I have been given a fundemanetal matrix $e^{At}$ and am asked to find $A$. It's kind of an odd question as we've mainly been calculating $e^{At}$ when given a certain matrix $A$. My question is how do I find $A$ and if you can always find $A$ for any fundamental matrix. Anyways this was given:

\begin{align} e^{At}&= \begin{bmatrix} 2e^{2t}-e^t & e^{2t}-e^t&e^t-e^{2t} \\ e^{2t}-e^t & 2e^{2t}-e^t&e^t-e^{2t} \\ 3e^{2t}-3e^t & 3e^{2t}-3e^t&3e^t-2^{2t} \\ \end{bmatrix} \end{align}

$\endgroup$
9
$\begingroup$

Hint: Take the derivative towards $t$ and plug in $t=0$.

$\endgroup$
  • $\begingroup$ Yes, that seems to work! Thanks. I'm assuming since $e^{At}=X(t)X(0)^{-1}$ for a fundamental solution $X(t)$ we can always find $A$. $\endgroup$ – J Dijkstra Jan 15 '18 at 0:30
  • $\begingroup$ \begin{align} A&= \begin{bmatrix} 3 & 1&-1 \\ 1 & 3&-1 \\ 3& 3&-1 \\ \end{bmatrix} \end{align} $\endgroup$ – Unknown x Jan 15 '18 at 1:46
  • $\begingroup$ can you please tell me? Am I calculating the correct matrix? $\endgroup$ – Unknown x Jan 15 '18 at 14:07
  • $\begingroup$ @ManeeshNarayanan Correct. (You can also take a look in the answers of odd numbered exercises.) $\endgroup$ – The Phenotype Jan 15 '18 at 17:10
  • $\begingroup$ which textbook? $\endgroup$ – Unknown x Jan 15 '18 at 17:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.